Building an Enneper surface from asymptotic curves using straight planar plywood laths
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- Опубликовано: 2 окт 2024
- This video is showing me and Isak Näslund building an Enneper surface from 1mm plywood. It is possible to build flat but the built in energy from the bending of the laths makes it taking it certain shape.
An Enneper surface is what is called a minimal surface in mathematics. From the principal curvature parameterization it is pretty straight forward getting the asymptotic lines on the surface.
mathworld.wolfr...
Utilizing the concept of asymptotic curves, which are surface curves whose osculating plane coincides with the tangent plane of the surface, the model could be built using planar straight laths of plywood.
You can read further about this in for instance Struik.
books.google.s...
Awesome work!
thanks! :)
Is this not an anticlastic shape - a hyperbolic paraboloid?
Hi! :)
You are correct, it is an anticlastic shape. Asymptotic lines only exist on anticlastic shapes. Though, the design was based on an Enneper surface because it is a minimal surface, meaning asymptotic curves intersect at right angles. We modified the parameterization following the principal lines by rotating the surface coordinates 45 degrees to follow the asymptotic curves. mathworld.wolfram.com/EnnepersMinimalSurface.html
If you look at the hyperbolic paraboloid, the mean curvature is not zero, thus not a minimal surface.
mathworld.wolfram.com/HyperbolicParaboloid.html
However, to ensure it would be an exact Enneper surface, we must force it to have exactly right angles in the intersections. Now, it finds the most "comfortable" shape. When we made a bigger structure, we used timber blocks at the intersections, which forced the laths to meet at right angles in the erected state ruclips.net/video/eCAMZNoaXYs/видео.html
Still, due to various factors such as self-weight, it tries to find a more comfortable shape which is likely slightly off the exact mathematical surface. For our purpose, this did not matter. I hope this answers your question :)
Wow, thats great, how u made it. Let us know the detailed tutorial pleeej
Hi, thanks for the comment, glad you find it interesting :) Maybe I can make a video of it in the future.
I do not know how much you know about math. However, for this case we used the principal curvature parameterization for an Enneper surface. That means that the coordinate curves follow principal curvature lines.
mathworld.wolfram.com/EnnepersMinimalSurface.html
On minimal surfaces, the asymptotic lines are easily found at 45 degrees from the principal curvature lines. Then we extruded the asymptotic lines from the surface to get the laths and made the slots such that we could assemble them. We used a laser cutter to cut out the laths in 3mm plywood.
I talk a bit about this kind of stuff in this video if you are interested:
ruclips.net/video/lybmJGUwR8s/видео.html
@@EmilAdiels Thanx a lot Sir, i m a college math teacher, also preparing math models , organizing math exhibitions and teaching math through different softwares as well. i watched the video through link shared. loved your work alot.
@@onlineskillsbyimranbilal Cool, sounds awesome! I think my own understanding and interest in math increased when I could visualize using different software's and started to code myself. Keep up the good work!
Glad you liked it!:)
Best wishes
Emil
Would you say that its possible to make this out of thicker paper as well?
Hi! Thanks for the comment!
I have not tried paper, but there might be a risk that the bending and the twist will make a permanent impact on the paper. Instead of a continuously bent lath it might break near the joints causing a lot of local bending around that area. However, I guess there is a great range of different types of paper that you can use, with different properties and quality.
Cheers
Emil
Hi Emil! !I'm currently working on a very similar project for school, but I'm struggling with the definition. Would you share your Grasshopper implementation for the asymptotic curve network?
Hi Dimitar,
Glad you find this interesting. It was a long time ago we did this project, so I don't know where the script is at the moment. I know we put some stuff up here for our different workshops. You might find something there:) github.com/archengtech
hi!! amazing work first of all! I am an architect student and want to try and build this model in a real-life scale! and have some questions. I will be glad if you could help me understand it better.
Hi Shannon,
Glad you find it exciting and also very sorry for my late reply.
You can find some information here in this conference paper we wrote:
www.researchgate.net/publication/336367443_The_design_fabrication_and_assembly_of_an_asymptotic_timber_gridshell
I also had a lecture where I discussed different aspects of an improved version of the grid shell described in the conference paper.
ruclips.net/video/lybmJGUwR8s/видео.html
I hope something might help you:)
I love this! I wish I had my family's mathematical genius. People make the neatest things out of his formula!
Thanks, I'm glad you liked it! :)
I had some difficulties with math when I was younger, but if you are interested and find it fun, you can improve a lot:)
@@EmilAdiels thank you! I do find math interesting. I have my first college class in mathematics next semester. I'm kind of nervous!
@@jenneper I understand. I was also pretty nervous before my first class, but I found it easier to team up with others to study. I often found that I learned as much from my fellow students as from the lectures. Most people are quite helpful and like to share their knowledge. That is at least my experience. Good luck!:)
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Amazing !
thanks!